Approximation of eigenvalues and eigenfunctions of the diffusion operator in a domain containing thin tubes by asymptotic domain decomposition method

• Published in: Applicable analysis pp. 1 - 23
• Main Authors: Amosov, Andrey, Gómez, Delfina, Panasenko, Grigory, Pérez-Martínez, Maria-Eugenia
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 11-07-2024
• Subjects: Mathematics
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036811.2024.2368699

We consider the spectral problem for the diffusion operator consideredin a domain containing thin tubes. A new version of the method of partialasymptotic decomposition of the domain is introduced to reduce thedimension inside the tubes, getting a model of hybrid dimensions. Themethod truncates the tubes at some small distance from the ends of thetubes and replaces the longer part of the tubes with segments. At the interfaceof the three-dimensional and one-dimensional subdomains, specialjunction conditions are set: the pointwise continuity of the flux and the continuityof the average over a cross-section of the eigenfunctions. We obtainconditions on the ratio of the characteristic sizes in the transverse and longitudinaldirections that ensure the closeness of two spectra, i.e. of thediffusion operator in the full-dimensional domain and the partially reducedone, keeping the conservation of the multiplicity, all up to a prescribedaccuracy.